User blog:Rgetar/Thoughts on further generalization of Veblen function
Argument of generalized Veblen function φ(X) is array of ordinals X, and its value is ordinal. X may be a natural number, then an ordinal, then, beginning from 11, a multi-ordinal array, then, beginning from 111, a multi-dimensional array of ordinals, then, beginning from 1111, a multi-trimensional array of ordinals, etc. To go further, we need larger arrays. Larger arrays Consider least array X such as X = X1. (Let me remind you how arrays are compared: first compared theirs elements with larger equal coordinates, then, if these elements are equal, elements with lesser equal coordinates, and so on). Let's denote this array as c0. So, c0 = c01 Least array larger than c0 is c01, 1. Then c01, 2 c01, ω c01, Ω c01, 11 c01, 111 c01, 1111 c02 c03 c0ω c01, 11 c01, 111 c01, 1111 c1 - second array X such as X = X1 c11, 1 c11, 11 c11, 111 c11, 1111 c11, c01 c11, c02 c11, c01, 11 c11, c01, 111 c11, c01, 1111 c12 c11, 11 c11, 111 c11, 1111 c2 c22 c21, 11 c21, 111 c21, 1111 c3 c4 c5 cω ... I think, that φ(c0) = BHO. And, I think, least ordinal such as α = φ(cα) we can designate as φ(c1, 0). So, subscript of c is array of ordinals, not just an ordinal. So, c1, 0 = c11 c111 c1111 cc0 ccc0 cccc0 ccccc0 d0 Arrays of arrays, Veblen-like functions I think, that we may designate these large arrays with Veblen-like function φ1(X1). Argument of φ1 is array of arrays of ordinals X1, and its value is array of ordinals (whereas argument of Veblen function φ is arrays of ordinals, and its value is ordinal). φ1(X1) is defined same way as φ(X), but if X1 is array of ordinals X, then φ1(X) = X1 (instead of φ(α) = ωα). To designate large arrays of arrays of ordinals we may similarly define another Veblen-like function φ2(X2), which argument is array of arrays of arrays of ordinals, and value is array of arrays of ordinals. And if X2 is array of arrays of ordinals X1, then φ2(X1) = ⟨1|X1⟩1 Then, to designate large arrays of arrays of arrays of ordinals, we may introduce another Veblen-like function φ3(X3), then φ4(X4), φ5(X5), and so on. To designate arrays of arrays we need new separators. I think, that we may use something like html tags: ⟨order|coordinates⟩element⟨/order⟩ where, say, for arrays of ordinals order = 0, for arrays of arrays of ordinals order = 1, for arrays of arrays of arrays of ordinals order = 2, and so on. For 0-th order: ⟨0|c⟩ = ⟨c⟩ ⟨/0⟩ = , So, ⟨0|c⟩e⟨/0⟩ = ⟨c⟩e, = ce, where e is ordinal, and c is array of ordinals. For 1-st order let's introduce some separator for ⟨/1⟩, for example, ;: ⟨/1⟩ = ; So, ⟨1|c⟩e⟨/1⟩ = ⟨1|c⟩e; where e is array of ordinal, and c is array of arrays of ordinals. 2-nd order: ⟨2|c⟩e⟨/2⟩ where e is array of arrays of ordinal, and c is array of arrays of arras of ordinals. Etc. A seperator (that is "closing tag") may be omitted, if this is last c-e pair in the array. So, φ1(0) = 01 = 1 φ1(1) = 11 φ1(1, 0) = φ1(11) = 111 φ1(111) = 1111 φ1(1111) = 11111 φ1(1; 0) = c0 φ1(c0) = c01 = c0 φ1(c01, 1) = c01, 11 φ1(c01, 11) = c01, 111 φ1(c01, 111) = c01, 1111 φ1(1; 1) = c1 φ1(1; 2) = c2 φ1(1; 3) = c3 φ1(1; ω) = cω φ1(1; 1, 0) = c1, 0 φ1(1; 111) = c111 φ1(2; 0) = d0 ... Numbers larger than any ordinal? Array 1, 0 is larger than any array α, but α is ordinal, so, arrays of ordinal may be considered as "extension" of ordinals. For example, 1, 0 may be considered as least number larger than any ordinal. So, the same way as ordinals may be considered as transfinite numbers, arrays of ordinals may be considered as "transordinal" numbers. Then, 1; 0 is larger than any array of ordinals, ⟨2|1⟩1⟨/2⟩ 0 is larger than any array of arrays of ordinals, ⟨3|1⟩1⟨/3⟩ 0 is larger than any array of arrays of arrays of ordinals, etc. But I suspect that this may be incorrect. Can object larger than any ordinal exist? Possibly, such an object also would be an ordinal, so, it cannot exist, but I'm not sure. Ordinals as arrays If arrays of ordinals are "transordinal" numbers, then, maybe, ordinals may be considered as arrays of natural numbers? Apparently, yes. So, 1, 0 is least ordinal larger than any natural number (that is ω), 1, 0, 0 is ω2, etc. In fact, this array is Cantor normal form, where + is separator, coefficients are elements, and exponents of ω are coordinates, that is ... ce, ... = ... ωce + ... And rule for Veblen function φ(X), X = α is similar to rule for Veblen-like function φ1(X1), X1 = X. Rule for Veblen-like function: φ1(X) = X1 Rule for Veblen function: φ(α) = α1 = ωα Arrays as uncountable ordinals If ordinals are arrays of natural numbers, then all elements of these arrays are less than ω. I'm still not sure, can numbers larger than any ordinal exist, and if not, then, maybe, arrays of ordinals are uncountable ordinals, not "transordinal" numbers? If so, then arrays of ordinals contain only elements lesser than first uncountable ordinal Ω, and 1, 0 = Ω. Then, array of arrays of ordinals is array of ordinals lesser than Ω2, and 1; 0 = Ω2. Arrays as positional numbering systems But why 1, 0 is Ω, not some other large ordinal, for example, Church-Kleene ordinal ω1CK, or Ωω? Maybe, this is free choice? If so, maybe, we can choose any ordinal α as limitation of array elements, so as all elements should not be larger than α? I think, yes. Then 1, 0 = α. For example, if α = 10 then 1, 0 = 10 1, 0, 0 = 100 etc. So, we get decimal numbering system. (Actually, I think, elements can be ≥ α, just 1, 0 = α. For example, in decimal system array 15, 11 is 15 * 10 + 11 * 1 = 161, but in this case different arrays can be equal: 15, 11 = 1, 6, 1). In general, it appears that array with limitation of elements α is α-ry positional numbering system. For example, for α = ω array is Cantor normal form, and, in fact, Cantor normal form is ω-ry numbering system. Array-ordinal correspondence Is it necessary to imply that an array of ordinals has "limitation of elements"? I think, if no, that is if arrays without limitation can exist, then "transordinal" numbers exist, and array 1, 0 without limitation is least "transordinal" number. And if yes, then we should imply that for given class of arrays 1, 0 is some ordinal. So, there is array-ordinal correspondence, and arrays of ordinals can be considered as ordinals. For example, array of Veblen function variables X. If we imply that for this array 1, 0 = Ω, then we can consider X as single ordinal written in Ω-ry positional numbering system. Example of Veblen function with X represented as single ordinal: φ(Ω) = ε0 φ(Ω + 1) = ε1 φ(Ω + ω) = εω φ(Ω2) = ζ0 φ(Ω3) = η0 φ(Ωω) = φ(ω, 0) φ(Ω2) = Γ0 φ(Ω23 + Ω5 + 7) = φ(3, 5, 7) φ(Ωω) = φ(ω1) = SVO φ(ΩΩ) = φ(Ω1, 0) = φ(Ω1) = φ(1, 01) = LVO ... Category:Blog posts